Decay Property for Solutions in the Three-Phase-Lag Heat Conduction

Abstract

In this paper, we consider the Cauchy problem of two models of the theory of heat conduction with three-phase-lag. Under appropriate assumptions on the material parameters, we show the optimal decay rate of the $L^2$norm of solutions. More precisely, we prove that in each model the $L^2$norm of the solution is decaying with the rate $(1+t)^{1/4}$ for initial data in $L^1(\mathbb{R})$. This decay rate is similar to the one of the heat kernel. Some faster decay rates have been also given for some weighted initial data in $L^1(\mathbb{R})$.